A potential deal is described as 'difficult', with Rodriguez in the midst of a two-year loan at Bayern Munich.
But it is claimed Liverpool have taken notice of the 26-year-old's displays, which have seen him score four times and provide five assists this season.
Rodriguez was signed in 2014 after a stunning World Cup for Colombia for 80 million Euros, a fee at the time equating to £63 million.
He was an instant hit, scoring 13 times and providing 13 assists in his first season, but he fell out of favour at Real Madrid after Zinedine Zidane's appointment.
Liverpool are said to be searching for a replacement for Philippe Coutinho, sold to Barcelona this month, and Rodriguez has the talent to fill the void.
This is certainly not the first time Rodriguez has been linked to Liverpool, but their need for him this time around is a little greater than previously thanks to Coutinho's departure.
A deal is certainly unlikely this January but in the summer the Reds may perhaps have a chance of persuading Real to try and break their agreement with Bayern if there is a clause allowing them to do so.
Here is a look at how Liverpool fans reacted on Twitter to the rumours...
If I were Jurgen Klopp I’d throw a cheeky bid at Real for James Rodriguez, would be a lovely bit of business, would likely get him for around half the Countinho transfer money. I can only dream though #LFC— Connah Selby (@ConnahSelby8) January 17, 2018
Liverpool now apparently interested in James Rodriguez I just want this transfer window to end already.— Jake (ITK) (@DancingFirmino) January 17, 2018
James Rodriguez would be a great fit taking over from Coutinho— Matt (@mattlfc6) January 17, 2018